On Directional Dependence in a Regression Line

Abstract

In this paper, under the assumption of linear relationship between two variables we provide alternative simple method of proving the existing result connecting correlation coefficient with those of skewness of response and explanatory variables. Further we have given a relationship between correlation coefficient and coefficient of kurtosis of response and explanatory variables assuming the linear relationship between the two variables. Simple alternative way of deriving the formula, which helps in finding the direction dependence in linear regression, is discussed.     

Exact expressions for the weights used in least-squares regression estimation for the log-logistic and Weibull distribution

Estimation for the log-logistic and Weibull distributions can be performed by using the equations used for probability plotting, and this technique outperforms the maximum likelihood (ML) estimation often in small samples. This leads to a highly heteroskedastic regression problem. Exact expressions for the variances of the residuals are derived which can be used to perform weighted regression. In large samples, the ML performs best, but it is shown that in smaller samples, the weighted regression outperforms the ML estimation with respect to bias and mean square error.     

A note on fourth moment-based direction dependence measures when regression errors are non normal

Measures of direction dependence enable researchers to determine the directionality of linear effects in bivariate data. Existing fourth moment-based approaches assume that regression errors are at least mesokurtic. Direction dependence measures based on the co-kurtosis of variables are proposed that relax this assumption. Simulations suggest that co-kurtosis-based measures perform equally well as existing kurtosis-based methods when distributional assumptions of the latter are fulfilled. However, kurtosis-based approaches are sensitive to platy- or leptokurtic errors, while co-kurtosis-based measures protect Type I error and power rates. Data requirements necessary for causal inference and recommendations for selecting proper direction dependence measures are discussed.     

Analysis of directional dependence using asymmetric copula-based regression models

The directional dependence between variables using asymmetric copula regression has drawn much attention in recent years. There are, however, some critical issues which have not been properly addressed in regards to the statistical inference of the directional dependence. For example, the previous use of asymmetric copulas failed to fully capture the dependence patterns between variables, and the method used for the parameter estimation was not optimal. In addition, no method was considered for selecting a suitable asymmetric copula or for computing the general measurements of the directional dependence when there are no closed-form expressions. In this paper, we propose a generalized multiple-step procedure for the full inference of the directional dependence in joint behaviour based on the asymmetric copula regression. The proposed procedure utilizes several novel methodologies that have not been considered in the literature of the analysis of directional dependence. The performance and advantages of the proposed procedure are illustrated using two real data examples, one from biological research on histone genes, and the other from developmental research on attention deficit hyperactivity disorder.     

Local Linear Least Squares Kernel Regression for Linear and Circular Predictors

We extend nonparametric regression models with local linear least squares fitting using kernel weights to the case of linear and circular predictors. We derive the asymptotic properties of the conditional bias and variance of bivariate local linear least squares kernel estimators. A small simulation study and a real experiment are given.     

A simulation study of five biased estimators for straight line regression

Five biased estimators of the slope in straight line regression are considered. For each, the estimate of the “bias parameter”, k, is a function of N, the number of observations, and [rcirc]² , the square of the least squares estimate of the standardized slope, β. The estimators include that of Farebrother, the ridge estimator of Hoerl, Kennard, and Baldwin, Vinod's shrunken estimators., and a new modification of one of the latter. Properties of the estimators are studied for 13 combinations of N and 3. Results of simulation experiments provide empirical evidence concerning the values of means and variances of the biased estimators of the slope and estimates of the “bias parameter”, the mean square errors of the estimators, and the frequency of improvement relative to least squares. Adjustments to degrees of freedom in the biased regression analysis of variance table are also considered. An extension of the new modification to the case of p> 1 independent variables is presented in an Appendix.     

Error variance estimation via least squares for small sample nonparametric regression

In this paper we explore statistical properties of some difference-based approaches to estimate an error variance for small sample based on nonparametric regression which satisfies Lipschitz condition. Our study is motivated by Tong and Wang (2005), who estimated error variance using a least squares approach. They considered the error variance as the intercept in a simple linear regression which was obtained from the expectation of their lag-k Rice estimator. Their variance estimators are highly dependent on the setting of a regressor and weight of their simple linear regression. Although this regressor and weight can be varied based on the characteristic of an unknown nonparametric mean function, Tong and Wang (2005) have used a fixed regressor and weight in a large sample and gave no indication of how to determine the regressor and the weight. In this paper, we propose a new approach via local quadratic approximation to determine this regressor and weight. Using our proposed regressor and weight, we estimate the error variance as the intercept of simple linear regression using both ordinary least squares and weighted least squares. Our approach applies to both small and large samples, while most existing difference-based methods are appropriate solely for large samples. We compare the performance of our approach with other existing approaches using extensive simulation study. The advantage of our approach is demonstrated using a real data set.     

Semiparametric Sieve-Type Generalized Least Squares Inference

This article considers the problem of statistical inference in linear regression models with dependent errors. A sieve-type generalized least squares (GLS) procedure is proposed based on an autoregressive approximation to the generating mechanism of the errors. The asymptotic properties of the sieve-type GLS estimator are established under general conditions, including mixingale-type conditions as well as conditions which allow for long-range dependence in the stochastic regressors and/or the errors. A Monte Carlo study examines the finite-sample properties of the method for testing regression hypotheses.     

Feasible robust estimator in restricted semiparametric regression models based on the LTS approach

Under some nonstochastic linear restrictions based on either additional information or prior knowledge in a semiparametric regression model, a family of feasible generalized robust estimators for the regression parameter is proposed. The least trimmed squares (LTS) method proposed by Rousseeuw as a highly robust regression estimator is a statistical technique for fitting a regression model based on the subset of h observations (out of n) whose least-square fit possesses the smallest sum of squared residuals. The coverage h may be set between n/2 and n. The LTS estimator involves computing the hyperplane that minimizes the sum of the smallest h squared residuals. For practical purpose, it is assumed that the covariance matrix of the error term is unknown and thus feasible estimators are replaced. Then, we develop an algorithm for the LTS estimator based on feasible methods. Through the Monte Carlo simulation studies and a real data example, performance of the feasible type of robust estimators is compared with the classical ones in restricted semiparametric regression models.     

A fuzzy robust regression approach applied to bedload transport data

Fuzzy least-square regression can be very sensitive to unusual data (e.g., outliers). In this article, we describe how to fit an alternative robust-regression estimator in fuzzy environment, which attempts to identify and ignore unusual data. The proposed approach concerns classical robust regression and estimation methods that are insensitive to outliers. In this regard, based on the least trimmed square estimation method, an estimation procedure is proposed for determining the coefficients of the fuzzy regression model for crisp input-fuzzy output data. The investigated fuzzy regression model is applied to bedload transport data forecasting suspended load by discharge based on a real world data. The accuracy of the proposed method is compared with the well-known fuzzy least-square regression model. The comparison results reveal that the fuzzy robust regression model performs better than the other models in suspended load estimation for the particular dataset. This comparison is done based on a similarity measure between fuzzy sets. The proposed model is general and can be used for modeling natural phenomena whose available observations are reported as imprecise rather than crisp.     

Optimal non-diagonal-type estimators in linear regression under the prediction error sum of squares criterion

This article considers the notion of the non-diagonal-type estimator (NDTE) under the prediction error sum of squares (PRESS) criterion. First, the optimal NDTE in the PRESS sense is derived theoretically and applied to the cosmetics sales data. Second, we make a further study to extend the NDTE to the general case of the covariance matrix of the model and then give a Bayesian explanation for this extension. Third, two remarks concerned with some potential shortcomings of the NDTE are presented and an alternative solution is provided and illustrated by means of simulations.     

Outlier detection using difference-based variance estimators in multiple regression

In this article, we propose an outlier detection approach in a multiple regression model using the properties of a difference-based variance estimator. This type of a difference-based variance estimator was originally used to estimate error variance in a non parametric regression model without estimating a non parametric function. This article first employed a difference-based error variance estimator to study the outlier detection problem in a multiple regression model. Our approach uses the leave-one-out type method based on difference-based error variance. The existing outlier detection approaches using the leave-one-out approach are highly affected by other outliers, while ours is not because our approach does not use the regression coefficient estimator. We compared our approach with several existing methods using a simulation study, suggesting the outperformance of our approach. The advantages of our approach are demonstrated using a real data application. Our approach can be extended to the non parametric regression model for outlier detection.     

Directional dependence via Gaussian copula beta regression model with asymmetric GARCH marginals

This article proposes a new directional dependence by using the Gaussian copula beta regression model. In particular, we consider an asymmetric Generalized AutoRegressive Conditional Heteroscedasticity (GARCH) model for the marginal distribution of standardized residuals to make data exhibiting conditionally heteroscedasticity to white noise process. With the simulated data generated by an asymmetric bivariate copula, we verify our proposed directional dependence method. For the multivariate direction dependence by using the Gaussian copula beta regression model, we employ a three-dimensional archemedian copula to generate trivariate data and then show the directional dependence for one random variable given two other random variables. With West Texas Intermediate Daily Price (WTI) and the Standard & Poor’s 500 (S&P 500), our proposed directional dependence by the Gaussian copula beta regression model reveals that the directional dependence from WTI to S&P 500 is greater than that from S&P 500 to WTI. To validate our empirical result, the Granger causality test is conducted, confirming the same result produced by our method.     

Mean square error matrix comparisons of estimators in linear regression

The aim of this paper is to provide criteria which allow to compare two estimators of the parameter vector in the linear regression model with respect to their mean square error matrices, where the main interest is focussed on the case when the difference of the covariance matrices is singular. The results obtained are applied to equality restricted and pretest estimators.     

Comment on a generalized stochastic restricted ridge regression estimator

In this note, we make some comments about the paper of Alheety and Kibria (2014 Alheety, M.I., Kibria, B.M.G. (2014). A generalized stochastic restricted ridge regression estimator. Commun. Stat. Theor. Meth. 43:4415–4427.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and correct the wrongly proved Theorems in that paper.     

A computational framework for variable selection in multivariate regression

Stepwise variable selection procedures are computationally inexpensive methods for constructing useful regression models for a single dependent variable. At each step a variable is entered into or deleted from the current model, based on the criterion of minimizing the error sum of squares (SSE). When there is more than one dependent variable, the situation is more complex. In this article we propose variable selection criteria for multivariate regression which generalize the univariate SSE criterion. Specifically, we suggest minimizing some function of the estimated error covariance matrix: the trace, the determinant, or the largest eigenvalue. The computations associated with these criteria may be burdensome. We develop a computational framework based on the use of the SWEEP operator which greatly reduces these calculations for stepwise variable selection in multivariate regression.     

Nonparametric regression for dependent data in the errors-in-variables problem

We consider the nonparametric estimation of the regression functions for dependent data. Suppose that the covariates are observed with additive errors in the data and we employ nonparametric deconvolution kernel techniques to estimate the regression functions in this paper. We investigate how the strength of time dependence affects the asymptotic properties of the local constant and linear estimators. We treat both short-range dependent and long-range dependent linear processes in a unified way and demonstrate that the long-range dependence (LRD) of the covariates affects the asymptotic properties of the nonparametric estimators as well as the LRD of regression errors does.     

Inference in segmented line regression: a simulation study

A segmented line regression model has been used to describe changes in cancer incidence and mortality trends [Kim, H.-J., Fay, M.P., Feuer, E.J. and Midthune, D.N., 2000, Permutation tests for joinpoint regression with applications to cancer rates. Statistics in Medicine, 19, 335–351. Kim, H.-J., Fay, M.P., Yu, B., Barrett., M.J. and Feuer, E.J., 2004, Comparability of segmented line regression models. Biometrics, 60, 1005–1014.]. The least squares fit can be obtained by using either the grid search method proposed by Lerman [Lerman, P.M., 1980, Fitting segmented regression models by grid search. Applied Statistics, 29, 77–84.] which is implemented in Joinpoint 3.0 available at http://srab.cancer.gov/joinpoint/index.html, or by using the continuous fitting algorithm proposed by Hudson [Hudson, D.J., 1966, Fitting segmented curves whose join points have to be estimated. Journal of the American Statistical Association, 61, 1097–1129.] which will be implemented in the next version of Joinpoint software. Following the least squares fitting of the model, inference on the parameters can be pursued by using the asymptotic results of Hinkley [Hinkley, D.V., 1971, Inference in two-phase regression. Journal of the American Statistical Association, 66, 736–743.] and Feder [Feder, P.I., 1975a, On asymptotic distribution theory in segmented regression Problems-Identified Case. The Annals of Statistics, 3, 49–83.] Feder [Feder, P.I., 1975b, The log likelihood ratio in segmented regression. The Annals of Statistics, 3, 84–97.] Via simulations, this paper empirically examines small sample behavior of these asymptotic results, studies how the two fitting methods, the grid search and the Hudson's algorithm affect these inferential procedures, and also assesses the robustness of the asymptotic inferential procedures.